Analyzing the Computational Complexity of Abstract Dialectical Frameworks via Approximation Fixpoint Theory

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Analyzing the Computational Complexity of Abstract Dialectical Frameworks via Approximation Fixpoint Theory

Hannes Strass

Computer Science Institute Leipzig University, Germany

Johannes Peter Wallner

Institute of Information Systems Vienna University of Technology, Austria

Abstract

Abstract dialectical frameworks (ADFs) have recently been proposed as a versatile generalization of Dung’s abstract argumentation frameworks (AFs). In this paper, we present a comprehensive analysis of the computational complexity of ADFs. Our results show that while ADFs are one level up in the polynomial hierarchy compared to AFs, there is a useful subclass of ADFs which is as complex as AFs while arguably offering more modeling capacities. As a technical vehicle, we employ the approximation fixpoint theory of Denecker, Marek and Truszczy´nski, thus showing that it is also a useful tool for complexity analysis of operator-based semantics.

Introduction

Formal models of argumentation are increasingly being re- cognized as viable tools in knowledge representation and reasoning (Bench-Capon and Dunne 2007). A particularly popular formalism are Dung’s abstract argumentation frameworks (AFs) (1995). AFs treat arguments as abstract entities and natively represent only attacks between them using a binary relation. Typically, abstract argumentation frameworks are used as a target language for translations from more concrete languages. For example, the Carneades formalism for structured argumentation (Gordon, Prakken, and Walton 2007) has been translated to AFs (Van Gijzel and Prakken 2011); Caminada and Amgoud (2007) and Wyner et al. (2013) translate rule-based defeasible theories into AFs. Despite their popularity, abstract argumentation frameworks have limitations. Most significantly, their lim- ited modeling capacities are a notable obstacle for applications: arguments can only attack one another. Furthermore, Caminada and Amgoud (2007) observed how AFs that arise as translations of defeasible theories sometimes lead to un- intuitive conclusions.

To address the limitations of abstract argumentation frameworks, researchers have proposed quite a number of generalizations of AFs (Brewka, Polberg, and Woltran 2013). Among the most general of those are Brewka and Woltran’s abstract dialectical frameworks (ADFs)(2010).

ADFs are even more abstract than AFs: while in AFs arguments are abstract and the relation between arguments is Copyright c2014, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved.

fixed to attack, in ADFs also the relations are abstract (and calledlinks). The relationship between different arguments (calledstatementsin ADFs) is specified byacceptance conditions. These are Boolean functions indicating the conditions under which a statementscan be accepted when given the acceptance status of all statements with a direct link tos (itsparents). ADFs have been successfully employed to address the shortcomings of AFs: Brewka and Gordon (2010) translated Carneades to ADFs and for the first time allowed cyclic dependencies amongst arguments; for rule-based defeasible theories we (Strass 2013b) showed how to deal with the problems observed by Caminada and Amgoud (2007).

There is a great number of semantics for AFs already, and many of them have been generalized to ADFs. Thus it might not be clear to potential ADF users which semantics are ad- equate for a particular application domain. In this regard, knowing the computational complexity of semantics can be a valuable guide. However, existing complexity results for ADFs are scattered over different papers, miss several semantics and some of them present upper bounds only. In this paper, we provide a comprehensive complexity analysis for ADFs. In line with the literature, we represent acceptance conditions by propositional formulas as they provide a compact and elegant way to represent Boolean functions.

Technically, we base our complexity analysis on the approximation fixpoint theory (AFT) by Denecker, Marek and Truszczy´nski (2000; 2003; 2004). This powerful framework provides an algebraic account of how monotone and nonmonotone two-valued operators can be approximated by monotone three- or four-valued operators. (As an example of an operator to be approximated, think of the two- valued van Emden-Kowalski consequence operator from logic programming.) AFT embodies the intuitions of decades of KR research; we believe that this is very valuable also for relatively recent languages (such as ADFs), because we get the enormously influential formalizations of intuitions of Reiter and others for free. (As a liberal variation on Newton, we could say that approximation fixpoint theory allows us to take the elevator up to the shoulders of giants instead of walking up the stairs.) In fact, approximation fixpoint theory can be and partially has already been used to define some of the semantics of ADFs (Brewka et al. 2013;

Strass 2013a). There, we generalized various AF and logic programming semantics to ADFs using AFT, which has

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provided us with two families of semantics, that we call – for reasons that will become clear later – approximate andultimate, respectively. Intuitively speaking, both families approximate the original two-valued model semantics of ADFs, where the ultimate family is moreprecisein a formally defined sense. The present paper employs approximating operators for complexity analysis and thus shows that AFT is also well-suited for studying the computational complexity of formalisms.

Along with providing a comparison of the approximate and ultimate families of semantics, our main results can be summarized as follows. We show that: (1) the computational complexity of ADF decision problems is one level up in the polynomial hierarchy from their AF counterparts (Dunne and Wooldridge 2009); (2) the ultimate semantics are as complex as the approximate semantics, with the notable ex- ception of two-valued stable models; (3) there is a certain subclass of ADFs, calledbipolarADFs (BADFs), which is of the same complexity as AFs. Intuitively, in bipolar ADFs all links between statements are supporting or attacking. To formalize these notions, Brewka and Woltran (2010) gave a precise semantical definition of support and attack. In our work, we assume that the link types are specified by the user along with the ADF. We consider this a harmless assumption since the existing applications of ADFs produce bipolar ADFs where the link types are known (Brewka and Gordon 2010; Strass 2013b). This attractiveness of bipolar ADFs from a KR point of view is the most significant result of the paper: it shows that BADFs offer – in addition to AF-like and more general notions of attack – also syntactical notions of supportwithout any increase in computational cost.

Previously, Brewka, Dunne and Woltran (2011) translated BADFs into AFs for two-valued semantics and suggested indirectly that the complexities align. Here we go a direct route, which has more practical relevance since it immediately affects algorithm design. Our work was also inspired by the complexity analysis of assumption-based argumentation by Dimopoulos, Nebel and Toni (2002) – they derived generic results in a way similar to ours.

The paper proceeds as follows. We first provide the background on approximation fixpoint theory, abstract dialectical frameworks and the necessary elements of complexity theory. In the section afterwards, we define the relevant decision problems, survey existing complexity results, use examples to illustrate how operators revise ADF interpretations and show generic upper complexity bounds. In the main section on complexity results for general ADFs, we back up the upper bounds with matching lower bounds; the section afterwards does the same for bipolar ADFs. We end with a brief discussion of related and future work. An earlier version of this paper with more details and all proofs is avail- able as a technical report (Strass and Wallner 2013).

Background

Acomplete lattice is a partially ordered set(A,v)where every subset of A has a least upper and a greatest lower bound. In particular, a complete lattice has a least and a greatest element. An operatorO:A→Aismonotoneif for allxvywe findO(x)vO(y). Anx∈Ais afixpointofO

ifO(x) =x; anx∈Ais aprefixpointofOifO(x)vxand apostfixpointofOifxvO(x). Due to a fundamental result by Tarski and Knaster, for any monotone operatorOon a complete lattice, the set of its fixpoints forms a complete lattice itself (Davey and Priestley 2002, Theorem 2.35). In particular, its least fixpointlfp(O)exists.

In this paper, we will be concerned with more general algebraic structures: complete partially ordered sets (CPOs).

A CPO is a partially ordered set with a least element where each directed subset has a least upper bound. A set is directed iff it is nonempty and each pair of elements has an upper bound in the set. Clearly every complete lattice is a complete partially ordered set, but not necessarily vice versa.

Fortunately, complete partially ordered sets still guarantee the existence of (least) fixpoints for monotone operators.

Theorem 1 ((Davey and Priestley 2002, Theorem 8.22)).

In a complete partially ordered set(A,v), anyv-monotone operatorO:A→Ahas a least fixpoint.

Approximation Fixpoint Theory

Denecker, Marek and Truszczy´nski (2000) introduce the im- portant concept of an approximation of an operator. In the study of semantics of knowledge representation formalisms, elements of lattices represent objects of interest. Operators on lattices transform such objects into others according to the contents of some knowledge base. Consequently, fixpoints of such operators are then objects that are fully up- dated – informally, the knowledge base can neither increase nor decrease the amount of information in a fixpoint.

To study fixpoints of operatorsO, DMT study theirap- proximation operatorsO. WhenOoperates on a setA, its approximationOoperates on pairs(x, y)∈A×A. Such a pair (x, y)can be seen as representing asetof lattice elements by providing a lower bound xand an upper bound y. Consequently, (x, y)approximates all z∈A such that xvzvy. We will restrict our attention toconsistentpairs – those where xvy, that is, the set of approximated elements is nonempty; we denote the set of all consistent pairs overAbyA^c. A pair(x, y)withx=yis calledexact– it

“approximates” a single element of the original lattice.

It is natural to order approximating pairs according to their information content. Formally, forx1, x2, y1, y2∈A define the information ordering (x1, y1)≤i(x2, y2) iff x1vx2 andy2vy1. This ordering and the restriction to consistent pairs leads to a complete partially ordered set (A^c,≤i), theconsistent CPO. For example, thetrivial pair (⊥,>)consisting ofv-least ⊥andv-greatest lattice element>approximates all lattice elements and thus contains no information – it is the least element of the CPO(A^c,≤i);

exact pairs(x, x)are the maximal elements of(A^c,≤_i).

To define an approximation operator O:A^c→A^c, one essentially has to define two functions: a function O⁰:A^c →Athat yields a revisedlowerbound (first component) for a given pair; and a function O⁰⁰:A^c →A that yields a revised upper bound (second component) for a given pair. Accordingly, the overall approximation is then given by O(x, y) = (O⁰(x, y),O⁰⁰(x, y)) for (x, y)∈A^c. The operatorO:A^c →A^c is approximating

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Kripke-Kleene semantics lfp(O) grounded pair admissible/reliable pair(x, y) (x, y)≤_i O(x, y) admissible pair three-valued supported model(x, y) (x, y) =O(x, y) complete pair M-supported model(x, y) (x, y)≤i O(x, y)and(x, y)is≤i-maximal preferred pair

two-valued supported model(x, x) (x, x) =O(x, x) model

two-valued stable model(x, x) x=lfp(O⁰(·, x)) stable model

Table 1: Operator-based semantical notions (and their argumentation names on the right) for a complete lattice(A,v)and an approximating operatorO:A^c→A^con the consistent CPO. While an approximating operator always possesses three-valued (post-)fixpoints, two-valued fixpoints need not exist. Clearly, any two-valued stable model is a two-valued supported model is a preferred pair is a complete pair is an admissible pair; furthermore the grounded semantics is a complete pair.

iff it is ≤i-monotone and it satisfies O⁰(x, x) =O⁰⁰(x, x) for allx∈A, that is, Oassigns exact pairs to exact pairs.

Such an O thenapproximates an operatorO:A→A on the original lattice iffO⁰(x, x) =O(x)for allx∈A.

The main contribution of Denecker, Marek and Truszczy´nski (2000) was the association of the stable operator to an approximating operator. Their original definition was four-valued; in this paper we are only interested in two-valued stable models and simplified the definitions. For an approximating operatorOon a consistent CPO, a (two-valued) pair(x, x)∈A^cis a (two-valued) stable model ofOiffxis the least fixpoint of the operator O⁰(·, x)defined byw7→ O⁰(w, x)forwvx. This general, lattice-theoretic approach yields a uniform treatment of the standard semantics of the major nonmonotonic knowledge representation formalisms – logic programming, default logic and autoepistemic logic (Denecker, Marek, and Truszczy´nski 2003).

In subsequent work, Denecker, Marek and Truszczy´nski (2004) presented a general, abstract way to define the most precise – called the ultimate – approximation of a given operator O. Most precise here refers to a generalisation of ≤i to operators, where for O1,O2, they define O1≤iO2 iff for all (x, y)∈A^c it holds that O1(x, y)≤iO2(x, y). Denecker, Marek and Truszczy´nski (2004) show that the most precise approximation ofOisU_O:A^c →A^cthat maps(x, y)to

l{O(z)|xvzvy},G

{O(z)|xvzvy}

whereudenotes the greatest lower bound and tthe least upper bound in the complete lattice(A,v).

In recent work, we defined new operator-based semantics inspired by semantics from logic programming and abstract argumentation (Strass 2013a).¹ An overview is in Table 1.

Abstract Dialectical Frameworks

An abstract dialectical framework (ADF) is a directed graph whose nodes represent statements or positions which can be accepted or not. The links represent dependencies: the status of a nodesonly depends on the status of its parents (denoted par(s)), that is, the nodes with a direct link tos. In addition,

1To be precise, we used a slightly different technical setting there. The results can however be transferred to the present setting (Denecker, Marek, and Truszczy´nski 2004, Theorem 4.2).

each nodeshas an associated acceptance conditionC_sspe- cifying the exact conditions under whichsis accepted. Cs

is a function assigning to each subset ofpar(s)one of the truth values t,f. Intuitively, if for someR ⊆ par(s) we haveC_s(R) =t, thenswill be accepted provided the nodes inRare accepted and those inpar(s)\Rare not accepted.

Definition 1. Anabstract dialectical frameworkis a tuple Ξ = (S, L, C)where

• Sis a set of statements (positions, nodes),

• L⊆S×Sis a set of links,

• C={Cs}s∈S is a collection of total functions Cs: 2^par^(s)→ {t,f}, one for each statement s. The functionC_sis calledacceptance condition ofs.

It is often convenient to represent acceptance conditions by propositional formulas. In particular, we will do so for the complexity results of this paper. There, each C_s is represented by a propositional formula ϕs overpar(s).

Then, clearly, C_s(R∩par(s)) =t iff R|=ϕ_s. Further- more, throughout the paper we will denote ADFs byΞand tacitly assume thatΞ = (S, L, C)unless stated otherwise.

Brewka and Woltran (2010) introduced a useful subclass of ADFs calledbipolar: Intuitively, in bipolar ADFs (BADFs) each link is supporting or attacking (or both).

Formally, a link(r, s)∈Lissupporting inΞiff for allR⊆ par(s), we have thatC_s(R) =timpliesC_s(R∪ {r}) =t;

symmetrically, a link (r, s) ∈ L isattacking in Ξ iff for all R ⊆ par(s), we have that C_s(R∪ {r}) = timplies Cs(R) =t. An ADFΞ = (S, L, C)isbipolariff all links inLare supporting or attacking or both; we useL⁺ to denote all supporting andL⁻to denote all attacking links ofL inΞ. For ans∈Swe defineattΞ(s) = {x|(x, s)∈L⁻} andsupp_Ξ(s) ={x|(x, s)∈L⁺}.

The semantics of ADFs can be defined using approximating operators. For two-valued semantics of ADFs we are interested in sets of statements, that is, we work in the complete lattice (A,v) = (2^S,⊆). To approximate elements of this lattice, we use consistent pairs of sets of statements and the associated consistent CPO(A^c,≤i)– theconsistent CPO overS-subset pairs. Such a pair(X, Y)∈A^c can be regarded as a three-valued interpretation where all elements inXare true, those inY\Xare unknown and those inS\Y are false. (This allows us to use “pair” and “interpretation”

synonymously from now on.) The following definition spe- cifies how to revise a given three-valued interpretation.

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Definition 2 ((Strass 2013a, Definition 3.1)). Let Ξ be an ADF. Define an operatorGΞ: 2^S×2^S →2^S×2^Sby

GΞ(X, Y) = (G_Ξ⁰(X, Y),G_Ξ⁰(Y, X))

G_Ξ⁰(X, Y) ={s∈S | B⊆par(s), C_s(B) =t, B⊆X, (par(s)\B)∩Y =∅}

Intuitively, statement s is included in the revised lower bound iff the input pair provides sufficient reason to do so, given acceptance condition Cs. Although the operator is defined for all pairs (including inconsistent ones), its restriction to consistent pairs is well-defined since it maps consistent pairs to consistent pairs. This operator defines the approximate family of ADF semantics according to Table 1. Based on the three-valued operatorGΞ, a two-valued one-step consequence operator for ADFs can be defined by GΞ(X) =G_Ξ⁰(X, X). The general result of Denecker, Marek and Truszczy´nski (2004) (Theorem 5.6) then immediately defines the ultimate approximation ofGΞas the oper- atorU_Ξgiven byU_Ξ(X, Y) = (U_Ξ⁰(X, Y),U_Ξ⁰⁰(X, Y))with

• U_Ξ⁰(X, Y) ={s∈S| for allX⊆Z ⊆Y, Z |=ϕs}and

• U_Ξ⁰⁰(X, Y) ={s∈S| for someX ⊆Z ⊆Y, Z|=ϕ_s}.

Incidentally, Brewka and Woltran (2010) already defined this operator, which was later used to define the ultimate family of ADF semantics according to Table 1 (Brewka et al. 2013).² In this paper, we will refer to the two families of three-valued semantics as “approximateσ” and “ultimateσ”

forσamong admissible, grounded, complete, preferred and stable. For two-valued supported models (or simply models), approximate and ultimate semantics coincide.

Finally, for a propositional formulaϕover vocabularyP and X⊆Y ⊆P we define the partial valuation of ϕ by (X, Y)asϕ^(X,Y⁾=ϕ[p/t:p∈X][p/f :p∈P\Y]. This partial evaluation takes the two-valued part of(X, Y)and replaces the evaluated variables by their truth values. Natur- ally,ϕ^(X,Y⁾is a formula over the vocabularyY \X. Complexity theory

We assume familiarity with the complexity classesP, NP andcoNP, as well as with polynomial reductions and hardness and completeness for these classes. We also make use of the polynomial hierarchy, that can be defined (using oracle Turing machines) as follows: Σ^P₀ = Π^P₀ = ∆^P₀ =P, Σ^P_i+1=NP^Σ^Pⁱ,Π^P_i+1=coNP^Σ^Pⁱ ,∆^P_i+1=P^Σ^Pⁱ fori≥0.

As a somewhat non-standard polynomial hierarchy complexity class, we use D^P_i , a generalisation of the complexity classDPto the polynomial hierarchy. A language is inDP iff it is the intersection of a language inNPand a language incoNP. Generally, a language is in D^P_i iff it is the intersection of a language inΣ^P_i and a language inΠ^P_i . The ca- nonical problem ofDP=D^P₁ is SAT-UNSAT, the problem to decide for a given pair(ψ₁, ψ₂)of propositional formulas whetherψ1is satisfiable andψ2is unsatisfiable. Obviously, by definitionΣ^P_i,Π^P_i ⊆D^P_i ⊆∆^P_i+1for alli≥0.

2Technically, Brewka et al. (2013) represented interpretations not by pairs(X, Y)∈A^cbut by mappingsv:S→ {t,f,u}into the set of truth valuest(true),f (false) andu(unknown or undecided). Clearly the two representations are interchangeable.

Preparatory Considerations

We first introduce some notation to make precise what decision problems we will analyze. For a setS, let

• (A^c,≤i)be the consistent CPO ofS-subset pairs,

• Oan approximating operator on(A^c,≤i),

• σ ∈ {adm, com, grd, pre,2su,2st}a semantics among admissible, complete, grounded, preferred, two-valued supported and two-valued stable semantics, respectively.

In theverificationproblem we decide whether(X, Y)∈A^c is aσ-model/pair ofO, denoted byVer_σÔ(X, Y). In theex- istenceproblem we ask whether there exists aσ-model/pair of O which is non-trivial, that is, different from (∅, S), denoted by ExistsÔ_σ. For query reasoning and s∈S we consider the problem of deciding whether there exists a σ-model/pair (X, Y) of O such that s∈X, denoted by CredÔ_σ(s)(credulousreasoning) and the problem of deciding whether in all σ-models/pairs (X, Y) of O we have s ∈ X, denoted bySkeptÔ_σ(s)(skepticalreasoning). Note that it is no restriction to check only for truth of a statement s∈S, since checking for falsity can be modeled by introdu- cing a new statements⁰that behaves like the logical negation ofs, by setting its acceptance condition toϕs⁰ =¬s.

Existing results

We briefly survey – to the best of our knowledge – all existing complexity results for abstract dialectical frameworks.

For general ADFsΞand the ultimate family of semantics, Brewka et al. (2013) have shown the following:

• Ver_2su^U^Ξ is inP,Exists^U_2su^Ξ isNP-complete (Proposition 5)

• Ver_adm^U^Ξ iscoNP-complete (Proposition 10)

• Ver_grd^U^Ξ andVer^U_com^Ξ areDP-complete (Theorem 6, Cor. 7)

• Ver_2st^U^Ξ is inDP (Proposition 8)

• Exists^U_2st^Ξ isΣ^P₂-complete (Theorem 9) For bipolar ADFs, Brewka and Woltran (2010) showed that Ver^U_grd^Ξ is inP(Proposition 15). So particularly for BADFs, this paper will greatly illuminate the complexity landscape.

Relationship between the operators

SinceUΞis the ultimate approximation ofGΞit is clear that for any X⊆Y ⊆S we have G_Ξ(X, Y)≤_iU_Ξ(X, Y). In other words, the ultimate revision operator produces new bounds that are at least as tight as those of the approximate operator. More explicitly, the ultimate new lower bound always contains the approximate new lower bound:

G_Ξ⁰(X, Y)⊆ U_Ξ⁰(X, Y); conversely, the ultimate new upper bound is contained in the approximate new upper bound:

U_Ξ⁰⁰(X, Y)⊆ G_Ξ⁰⁰(X, Y). Somewhat surprisingly, it turns out that the revision operators for the upper bound coincide.

Lemma 2. LetΞ = (S, L, C)be an ADF andX ⊆Y ⊆S.

G_Ξ⁰⁰(X, Y) =U_Ξ⁰⁰(X, Y)

The operators for computing a new lower bound are demonstrably different, since we can findΞand(X, Y)with U_Ξ⁰(X, Y)6⊆ G_Ξ⁰(X, Y), as the following ADF shows.

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Example 1. Consider the ADFD= ({a},{(a, a)},{ϕa}) with one self-dependent statementathat has acceptance for- mulaϕ_a=a∨ ¬a. In Figure 1, we show the relevant CPO and the behavior of approximate and ultimate operators:

we see thatG_D(∅,{a})<_iU_D(∅,{a}), which shows that in some cases the ultimate operator is strictly more precise.

So in a sense the approximate operator cannot see beyond the case distinctiona∨¬a. As we will see shortly, this difference really amounts to the capability of tautology checking.

Example 2. ADF E= ({a, b},{(b, a),(b, b)},{ϕa, ϕb}) has acceptance formulasϕ_a=b∨ ¬b andϕ_b=¬b. Sob is self-attacking and the link fromb toais redundant. In Figure 1 on the next page, we show the relevant CPO and the behavior of the operatorsUEandGEon this CPO.

The examples show that the approximate and ultimate families of semantics really are different, save for one straightforward inclusion relation in case of admissible.

Corollary 3. For any ADFΞ, we have the following:

1. An approximate admissible pair is an ultimate admissible pair, but not vice versa.

2. With respect to their sets of pairs, the approximate and ultimate versions of preferred/complete/grounded semantics are⊆-incomparable.

Operator complexities

We next analyze the computational complexity of deciding whether a single statement is contained in the lower or upper bound of the revision of a given pair. This then leads to the complexity of checking whether current lower/upper bounds are pre- or postfixpoints of the revision operators for computing new lower/upper bounds, that is, whether the revisions represent improvements in terms of the information ordering. Intuitively, these results describe how hard it is to

“use” the operators and lay the foundation for the rest of the complexity results.

Proposition 4. LetΞbe an ADF,s∈SandX⊆Y ⊆S.

1. Decidings∈ G_Ξ⁰(X, Y)is inP.

2. DecidingG_Ξ⁰(X, Y)⊆Xis inP.

3. DecidingX ⊆ G_Ξ⁰(X, Y)is inP.

Now letO ∈ {GΞ,UΞ}.

4. Decidings∈ O⁰⁰(X, Y)isNP-complete.

5. DecidingO⁰⁰(X, Y)⊆Y iscoNP-complete.

6. DecidingY ⊆ O⁰⁰(X, Y)isNP-complete.

Proof. We only show 1 and 4 as the rest follows suit.

1. SinceX ⊆Y, we have that whenever there exists aB ⊆ X∩par(s)withC_s(B) =tand(par(s)\B)∩Y =∅, we know thatB =X∩par(s). Thuss ∈ G_Ξ⁰(X, Y)iff C_s(X∩par(s)) =tand(par(s)\X)∩Y =∅. For acceptance functions represented by propositional formulas, both conditions can be checked in polynomial time.

4. Decidings∈ G_Ξ⁰⁰(X, Y)isNP-complete:

inNP: By definition,G_Ξ⁰⁰(X, Y) = G_Ξ⁰(Y, X). To verify s ∈ G_Ξ⁰(Y, X), we can guess a setM ⊆S and verify thatM ⊆Y,par(s)\M ⊆S\XandM |=ϕs.

NP-hard: For hardness, we provide a reduction from SAT. Letψbe a propositional formula over vocabulary P. Define an ADFΞ = (S, L, C)withS =P ∪ {z}

wherez /∈ P, ϕz = ψandϕp = pfor allp ∈ P.

Observe thatpar(z) =P, and setX =∅andY =P.

Nowz ∈ G_Ξ⁰⁰(X, Y)iffz∈ G_Ξ⁰(Y, X)iffz ∈ G_Ξ⁰(P,∅) iff there is anM ⊆PwithP\M∩∅=∅andM |=ϕ_z iff there is anM ⊆PwithM |=ψiffψis satisfiable.

These results can also be formulated in terms of partial evaluations of acceptance formulas: We haves∈ G_Ξ⁰(X, Y) iff the partial evaluation ϕ^(X,Ys ⁾ is a formula without variables that has truth value t. Similarly, we have s∈ G_Ξ⁰⁰(X, Y) iff the partial evaluation ϕ^(X,Ys ⁾ is satisfiable. Under standard complexity assumptions, computing a new lower bound with the ultimate operator is harder than with the approximate operator. This is because, intuitively, s∈ U_Ξ⁰(X, Y)iff the partial evaluationϕ^(X,Ys ⁾is a tautology.

Proposition 5. LetΞbe an ADF,s∈SandX ⊆Y ⊆S.

1. Decidings∈ U_Ξ⁰(X, Y)iscoNP-complete.

2. DecidingU_Ξ⁰(X, Y)⊆XisNP-complete.

3. DecidingX⊆ U_Ξ⁰(X, Y)iscoNP-complete.

Proof. We only show the first item since the remaining proofs work along the same lines. The hardness proof uses the ADF from Proposition 4.

incoNP: To decide thats /∈ U_Ξ⁰(X, Y), we guess aZwith X ⊆Z⊆Y and verify thatZ 6|=ϕs.

coNP-hard: SetX=∅andY =P. Nowz∈ U_Ξ⁰(X, Y)iff z ∈ U_Ξ⁰(∅, P)iff for allZ ⊆P, we haveZ |=ϕ_z iff for allZ⊆P, we haveZ|=ψiffψis a tautology.

The next result considerably simplifies the complexity analysis of deciding the existence of non-trivial pairs.

Lemma 6. Let(A,v)be a complete lattice andO an approximating operator onA^c. The following are equivalent:

1. Ohas a non-trivial admissible pair.

2. Ohas a non-trivial preferred pair.

3. Ohas a non-trivial complete pair.

Proof. “(1)⇒(2)”: Let (⊥,>) <i (x, y) ≤i O(x, y).

We show that there is a preferred pair (p, q) ≥i

(x, y). Define D={(a, b)|(x, y)≤i(a, b)}, then the pair (D,≤_i)is a CPO on whichO is an approximating operator. (Obviously(a, b)∈D implies(x, y)≤i(a, b) whence by presumption and ≤_i-monotonicity of O we get(x, y)≤iO(x, y)≤iO(a, b)andO(a, b)∈D.) Now any sequence(a, b)≤i O(a, b)≤iO(O(a, b))≤i. . .is a non-empty chain inD and therefore has an upper bound inD. By Zorn’s lemma, the set of allO-admissible pairs inAhas a maximal element(p, q)≥i (x, y)>_i(⊥,>).

“(2)⇒(3)”: By (Strass 2013a, Theorem 3.10), every preferred pair is complete.

“(3)⇒(1)”: Any complete pair is admissible (Table 1).

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operator visualization:

approximate ultimate both

(∅,{a})

(∅,∅) ({a},{a})

(∅,{a, b}) (∅,{b})

(∅,{a}) ({a},{a, b}) ({b},{a, b})

(∅,∅) ({a},{a}) ({b},{b}) ({a, b},{a, b})

Figure 1: Hasse diagrams of consistent CPOs for the ADFs from Example 1 (left) and Example 2 (right). Solid lines represent the information ordering≤i. Directed arrows express how revision operators map pairs to other pairs. For pairs where the revisions coincide, the arrows are densely dashed andviolet. When the operators revise a pair differently, we use a dotted redarrow for the ultimate and a loosely dashedbluearrow for the approximate operator. Exact (two-valued) pairs are the≤_i- maximal elements. For those pairs, (and any ADFΞ) it is clear that the operatorsUΞandGΞcoincide since they approximate the same two-valued operatorG_Ξ. In Example 1 on the left, we can see that the ultimate operator maps all pairs to its only fixpoint({a},{a})whereais true. The approximate operator has an additional fixpoint,(∅,{a}), whereais unknown. In Example 2 on the right, the major difference between the operators is whether statementacan be derived given thatbhas truth value unknown. This is the case for the ultimate, but not for the approximate operator. Since there is no fixpoint in the upper row (showing the two-valued operatorGE), the ADFEdoes not have a two-valued model. Each of the revision operators has however exactly one three-valued fixpoint, which thus constitutes the respective grounded, preferred and complete semantics.

This directly shows the equivalence of the respective decision problems, that is,ExistsÔ_adm=ExistsÔ_pre=ExistsÔ_com.

Regarding decision problems for querying, skeptical reasoning w.r.t. admissibility is trivial, i.e.(∅, S)is always an admissible pair in any ADF. Further credulous reasoning w.r.t.

admissibility, complete and preferred semantics coincides.

Lemma 7. Let Ξ be an ADF, O ∈ {G_Ξ,U_Ξ} and s∈S.

ThenCredÔ_adm(s)iffCredÔ_com(s)iffCredÔ_pre(s).

Proof. Assume(X, Y)withs ∈ X is admissible w.r.t.O, then there exists a(X⁰, Y⁰)with(X, Y)≤i(X⁰, Y⁰)which is preferred with respect toOand wheres∈X⁰, see proof of Lemma 6. Since any preferred pair is also complete and any complete pair is also admissible the claim follows.

Generic upper bounds

We now show generic upper bounds for the computational complexity of the considered problems. This kind of analysis is in the spirit of the results of Dimopoulos, Nebel and Toni (2002) (Section 4). The first item is furthermore a straightforward generalization of Theorem 6.13 in (De- necker, Marek, and Truszczy´nski 2004).

Theorem 8. Let S be a finite set, define A= 2^S and let O be an approximating operator on (A^c,≤i), the consistent CPO ofS-subset pairs. For a pair(X, Y)∈A^c and an s∈S, let the problem of deciding whethers∈ O⁰(X, Y)be

inΠ^P_i ; let the problem of decidings∈ O⁰⁰(X, Y)be inΣ^P_i . For(X, Y)∈A^cand a statements∈S, we have:

1. The least fixpoint ofOcan be computed in polynomial time with a polynomial number of calls to aΣ^P_i-oracle.

2. Ver_admÔ (X, Y)is inΠ^P_i ;CredÔ_adm(s)is inΣ^P_i+1; 3. Ver_comÔ (X, Y)is inD^P_i ;CredÔ_com(s)is inΣ^P_i+1;

4. Ver_preÔ(X, Y) is in Π^P_i+1; CredÔ_pre(s) is in Σ^P_i+1; SkeptÔ_pre(s)is inΠ^P_i+2.

Proof.1. For any (V, W)∈A^c we can use the oracle to compute an application ofO⁰ by simply asking whether z∈ O⁰(V, W) for each z∈S. This means we can compute with a linear number of oracle calls the sets O⁰(V, W)andO⁰⁰(V, W), thus the pairO(V, W). Hence we can compute the sequence (∅, S) ≤i O(∅, S) ≤i

O(O(∅, S))≤i . . .which converges to the least fixpoint ofOafter a linear number of operator applications.

2. We can decide O⁰(X, Y)⊆X and Y ⊆ O⁰⁰(X, Y) in Σ^P_i , X ⊆ O⁰(X, Y) andO⁰⁰(X, Y)⊆Y inΠ^P_i ; all by combining independent guesses. Then Ver^O_adm(X, Y)is inΠ^P_i . For Cred^O_adm(s), we guess a pair(X1, Y1)with s∈X1and check if it is admissible.

3. Ver_comÔ (X, Y)is in D^P_i by the same method as for admissibility.CredÔ_com(s) =CredÔ_adm(s)by Lemma 7.

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4. ForVer_preÔ(X, Y), consider the co-problem, i.e. deciding whether(X, Y)is not a preferred pair. We first check if (X, Y)is a complete pair, which is in D^P_i . If this holds, we guess an(X1, Y1)with(X, Y)<i(X1, Y1)and check if it is complete.CredÔ_pre(X, Y): coincides with credulous reasoning w.r.t. admissibility, see Lemma 7. SkeptÔ_pre(s):

Consider the co-problem, i.e. deciding whether there exists a preferred pair(X₁, Y₁)withX₁∩ {a} = ∅. We guess such a pair(X1, Y1)and check if it is preferred.

Naturally, the capability of solving the functional problem ofcomputingthe grounded semantics allows us to solve the associated decision problems.

Corollary 9. Under the assumptions of Theorem 8, the problemsVer^O_grdandExists^O_grdare in∆^P_i+1.

Complexity of General ADFs

Due to the coincidence of G_Ξ⁰⁰ andU_Ξ⁰⁰, the computational complexities of decision problems that concern only the upper bound operator also coincide. This will save both work and space in the subsequent developments. Additionally, for all containment results (except for the grounded semantics), we can use Theorem 8 and need only show hardness.

Proposition 10. LetΞbe an ADF,X, Y ⊆Sand consider anyO ∈ {GΞ,UΞ}.Ver_adm^O (X, Y)iscoNP-complete.

Proof. Hardness follows from Proposition 4, item 5.

Recall that a pair(X, Y)is an approximate/ultimate complete pair iff it is a fixpoint of the corresponding (approximate/ultimate) operator. Given the complexities of operator computation, it is straightforward to show the following.

Proposition 11. LetΞ be an ADF, X⊆Y ⊆S and consider anyO ∈ {G_Ξ,U_Ξ}.Ver^O_com(X, Y)isDP-complete.

Next, we analyze the complexity of verifying that a given pair is the approximate (ultimate) Kripke-Kleene semantics of an ADFΞ, that is, the least fixpoint ofG_Ξ(U_Ξ). Although interesting, the proof is lengthy and technical, so we unfortu- nately have to omit it due to space constraints. Interestingly, the membership part is the tricky one, where we encode the steps of the operator computation into propositional logic.

Theorem 12. LetΞbe an ADF,X ⊆Y ⊆S and consider any operatorO ∈ {GΞ,UΞ}.Ver^O_grd(X, Y)isDP-complete.

We next ask whether there exists anon-trivialadmissible pair, that is, if at least one statement has a truth value other than unknown. Clearly, we can guess a pair and perform the coNP-check to show that it is admissible. The next result shows that this is also the best we can do. Again, the proof is lengthy and technical and we could not include it here.

Theorem 13. LetΞbe an ADF and consider any operator O ∈ {GΞ,UΞ}.Exists^O_admisΣ^P₂-complete.

Lemma 6 implies the same complexity for the existence of non-trivial complete and preferred pairs.

Corollary 14. LetΞbe an ADF,σ∈ {com, pre}and consider any operatorO ∈ {GΞ,UΞ}.Exists^O_σ isΣ^P₂-complete.

By corollary to Theorem 12, the existence of a non-trivial grounded pair can be decided inDPby testing whether the trivial pair(∅, S)is (not) a fixpoint of the relevant operator.

The following result shows that this bound can be improved.

We cannot present the proof here but can say that intuitively, half of the usual subset checks can be left out due to using the trivial pair.

Proposition 15. LetΞbe an ADF and consider any operator O ∈ {GΞ,UΞ}.Exists^O_grdiscoNP-complete.

Using the result for existence of non-trivial admissible pairs, the verification complexity for the preferred semantics is straightforward to obtain, similarly as in the case of AFs (Dimopoulos and Torres 1996).

Proposition 16. LetΞ be an ADF, X ⊆Y ⊆S and consider anyO ∈ {GΞ,UΞ}.Ver_pre^O(X, Y)isΠ^P₂-complete.

Considering query reasoning we now show that on general ADFs credulous reasoning with respect to admissibility is harder than on AFs. By Lemma 7, the same lower bound holds for complete and preferred semantics.

Proposition 17. LetΞbe an ADF,O ∈ {GΞ,UΞ}be an operator ands∈S.Cred^O_adm(s)isΣ^P₂-complete.

For credulous and skeptical reasoning with respect to the grounded semantics, we first observe that the two coincide since there is always a unique grounded pair. Furthermore, a statementsis true in the approximate grounded pair iffsis true in the least fixpoint (ofGΞ) iffsis true in all fixpoints iff there is no fixpoint wheresis unknown or false. This condition can be encoded in propositional logic and leads to the next result. For the ultimate operator we can use results for the verification problem (Brewka et al. 2013, Theorem 6).

Briefly put, the problem is incoNPsince theNP-hardness comes from verifying that certain arguments are undefined in the ultimate grounded pair, which is not needed for credulous reasoning. ForcoNP-hardness the proof of (Brewka and Woltran 2010, Proposition 13) can be easily adapted.

Proposition 18. Let Ξ be an ADF, O ∈ {G_Ξ,U_Ξ}, s∈S.

BothCred^O_grd(s)andSkept^O_grd(s)arecoNP-complete.

Regarding skeptical reasoning for the remaining semantics we need only show the results for complete and preferred semantics, in all other cases the complexity coincides with credulous reasoning or is trivial. For complete semantics it is easy to see that skeptical reasoning coincides with skeptical reasoning under grounded semantics, since the grounded pair is the≤i-least complete pair.

Corollary 19. LetΞbe an ADF,O ∈ {GΞ,UΞ}ands∈S.

Skept^O_com(s)iscoNP-complete.

Similar to reasoning on AFs, we step up one level of the polynomial hierarchy by changing from credulous to skeptical reasoning with respect to preferred semantics, which makes skeptical reasoning under preferred semantics particularly hard. We apply proof ideas by (Dunne and Bench- Capon 2002) to proveΠ^P₃-hardness.

Theorem 20. LetΞbe an ADF,O ∈ {GΞ,UΞ}ands∈S.

Skept^O_pre(s)isΠ^P₃-complete.

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Two-valued semantics

The complexity results we have obtained so far might lead the reader to ask why we bother with the approximate oper- atorGΞat all: the ultimate operatorUΞis at least as precise and for all the semantics considered up to now, it has the same computational costs. We now show that for the verification of two-valued stable models, the operator for the upper bound plays no role and therefore the complexity difference between the lower bound operators for approximate (inP) and ultimate (coNP-hard) semantics comes to bear.

For the ultimate two-valued stable semantics, Brewka et al. (2013) already have some complexity results: model verification is in DP (Proposition 8), and model existence is Σ^P₂-complete (Theorem 9). We will show next that we can do better for the approximate version.

Proposition 21. LetΞbe an ADF andX ⊆Y ⊆S. Veri- fying thatX is the least fixpoint ofG_Ξ⁰(·, Y)is inP.

Proof sketch. Roughly, we construct the sequence defined by X0=∅ and Xi+1=G_Ξ⁰(Xi, Y) for i≥0, as long as Xi ⊆Y. By≤i-monotonicity ofGΞ, this sequence is mono- tonically⊆-increasing and so the procedure terminates after a linear number of steps. We then check ifXi+1=Xi=X, that is, the right fixpoint was reached.

In particular, the procedure can decide whetherY is the least fixpoint ofG_Ξ⁰(·, Y), that is, whether(Y, Y)is a two- valued stable model ofGΞ. This yields the next result.

Theorem 22. Let Ξ be an ADF and X⊆S.

1.Ver^G_2st^Ξ(X, X)is inP. 2.Exists^G_2st^Ξ isNP-complete.

The hardness direction of the second part is clear since the respective result from stable semantics of abstract argumentation frameworks carries over.

Brewka et al. (2013) showed thatVer^U_2st^Ξ is inDP(Propos- ition 8). As one of the most surprising results of this paper, we can improve that upper bound tocoNP. The proof is not at all trivial, but basically the operator for the upper bound (contributing theNPpart) is not really needed. Using the complexity of the lower revision operatorU_Ξ⁰, we can even show completeness forcoNP.

Proposition 23. Let Ξ be an ADF and X⊆S.

Ver_2st^U^Ξ(X, X)iscoNP-complete.

We now turn to the credulous and skeptical reasoning problems for the two-valued semantics. We first recall that a two-valued pair(X, X)is a supported model (or model for short) of an ADFΞiffG_Ξ(X, X) = (X, X). Thus it could equally well be characterized by the two-valued operator by saying thatXis a model iffGΞ(X) = X. Now sinceUΞis the ultimate approximation ofGΞ, alsoUΞ(X, X) = (X, X) in this case. Rounding up, this recalls that approximate and ultimate two-valued supported models coincide. Hence we get the following results for reasoning with this semantics.

Corollary 24. LetΞbe an ADF,O ∈ {GΞ,UΞ}be an operator and s∈S. The problemCred^O_2su(s) isNP-complete;

Skept^O_2su(s)iscoNP-complete.

For the approximate two-valued stable semantics, the fact that model verification can be decided in polynomial time leads to the next result.

O GΞ,UΞ GΞ,UΞ GΞ UΞ

σ adm com pre grd 2su 2st 2st

Ver^O_σ coNP-c DP-c Π^P₂-c DP-c inP coNP-c Exists^O_σ Σ^P₂-c Σ^P₂-c Σ^P₂-c coNP-c NP-c Σ^P₂-c

Cred^O_σ Σ^P₂-c Σ^P₂-c Σ^P₂-c coNP-c NP-c Σ^P₂-c Skept^O_σ trivial coNP-c Π^P₃-c coNP-c coNP-c Π^P₂-c

Table 2: Complexity results for semantics of ADFs.

Corollary 25. LetΞbe an ADF and s∈S. Cred^G_2st^Ξ(s)is NP-complete;Skept^G_2st^Ξ(s)iscoNP-complete.

For the ultimate two-valued stable semantics, things are bit more complex. The hardness reduction in the proof of Theorem 9 in (Brewka et al. 2013) makes use of a statement zthat is false in any ultimate two-valued stable model. This can be used to show the same hardness for the credulous reasoning problem for this semantics: we introduce a new statementxthat behaves just like¬z, thenxis true in some model if and only if there exists a model.

Proposition 26. LetΞbe an ADF ands∈S. The problem Cred^U_2st^Ξ(s)isΣ^P₂-complete.

A similar argument works for the skeptical reasoning problem: Given a QBF∀P∃Qψ, we construct its negation

∃P∀Q¬ψ, whose associated ADFDhas an ultimate two- valued stable model (wherezis false) iff∃P∀Q¬ψis true iff the original QBF∀P∃Qψis false. Hence∀P∃Qψis true iffzis true in all ultimate two-valued stable models ofD.

Proposition 27. LetΞbe an ADF ands∈S. The problem Skept^U_2st^Ξ(s)isΠ^P₂-complete.

Complexity of Bipolar ADFs

We first note that since BADFs are a subclass of ADFs, all membership results from the previous section immediately carry over. However, we can show that many problems will in fact become easier. Intuitively, computing the revision operators is now P-easy because the associated satisfiabil- ity/tautology problems only have to treat restricted acceptance formulas. In bipolar ADFs, by definition, if in some three-valued pair(X, Y)a statementsis accepted by a revision operator (s∈ O⁰(X, Y)), it will stay so if we set its undecided supporters to true and its undecided attackers to false. Symmetrically, if a statement is rejected by an operator (s /∈ O⁰⁰(X, Y)), it will stay so if we set its undecided supporters to false and its undecided attackers to true. This is the key idea underlying the next result.

Proposition 28. Let Ξ be a BADF with L=L⁺∪L⁻, O ∈ {GΞ,UΞ},s∈SandX⊆Y ⊆S.

1. Decidings∈ O⁰(X, Y)is inP.

2. Decidings∈ O⁰⁰(X, Y)is inP.

Using the generic upper bounds of Theorem 8, it is now straightforward to show membership results for BADFs with known link types.

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Corollary 29. Let Ξ be a BADF with L=L⁺∪L⁻, consider any operator O ∈ {GΞ,UΞ} and semantics σ∈ {adm, com}. ForX ⊆Y ⊆Sands∈S, we find that

• Ver^O_σ(X, Y)andVer_grd^O (X, Y)are inP;

• Ver^O_pre(X, Y)is incoNP;

• ExistsÔ_σ,ExistsÔ_pre,CredÔ_σ(s)andCredÔ_pre(s)are inNP;

• ExistsÔ_grd,CredÔ_grd(s),SkeptÔ_grd(s),SkeptÔ_com(s)are inP;

• Skept^O_pre(s)is inΠ^P₂.

Proof. Membership is due to Theorem 8 and the complexity bounds of the operators in BADFs in Proposition 28, just note thatΣ^P₀ = Π^P₀ =P. Ver^O_grd(X, Y)is inP^P = Pby Corollary 9. For the existence of non-trivial pairs we can simply guess and check in polynomial time for admissible pairs and thus also for complete and preferred semantics.

Hardness results straightforwardly carry over from AFs.

Proposition 30. Let Ξ be a BADF with L=L⁺∪L⁻, consider any operator O ∈ {GΞ,UΞ} and semantics σ∈ {adm, com, pre}. ForX ⊆Y ⊆Sands∈S:

• Ver^O_pre(X, Y)iscoNP-hard;

• Exists^O_σ andCred^O_σ(s)areNP-hard;

• Skept^O_pre(s)isΠ^P₂-hard.

Proof. Hardness results from AFs for these problems carry over to BADFs as for all semantics AFs are a special case of BADFs (Brewka et al. 2013; Strass 2013a). The complexities of the problems on AFs for admissible and preferred semantics are shown by (Dimopoulos and Torres 1996), except for the Π^P₂-completeness result of skeptical preferred semantics, which is shown by (Dunne and Bench- Capon 2002). The complete semantics is studied by (Coste-

Marquis, Devred, and Marquis 2005).

We next show that there is no hope that the existence problems for approximate and ultimate two-valued stable models coincide as there are cases when the semantics differ.

Example 3. Consider the BADFF = (S, L, C)with statements S={a, b, c} and acceptance formulas ϕ_a =t, ϕb=a∨c and ϕc=a∨b. The only two-valued supported model is (S, S) where all statements are true.

This pair is also an ultimate two-valued stable model, since U_F⁰(∅, S) ={a}, and both ϕ^({a},S)_b =t∨c and ϕ^({a},S)c =t∨bare tautologies, whence U_F⁰({a}, S) =S.

However, (S, S) is not an approximate two-valued stable model: although G_F⁰(∅, S) ={a}, then G_F⁰({a}, S) ={a}

and we thus cannot reconstruct the upper boundS. ThusF has no approximate two-valued stable models.

So approximate and ultimate two-valued stable model semantics are indeed different. However, we can show that the respective existence problems have the same complexity.

Proposition 31. Let Ξ be a BADF with L=L⁺∪L⁻, O ∈ {G_Ξ,U_Ξ} an operator and semantics σ∈ {2su,2st}.

ForX ⊆S,Ver^O_σ(X, X)is inP;Exists^O_σ isNP-complete.

σ adm com pre grd 2su 2st

Ver_σ^O inP inP coNP-c inP inP inP Exists^O_σ NP-c NP-c NP-c inP NP-c NP-c

Cred^O_σ NP-c NP-c NP-c inP NP-c NP-c Skept^O_σ trivial inP Π^P₂-c inP coNP-c coNP-c

Table 3: Complexity results for semantics of bipolar Ab- stract Dialectical Frameworks forO ∈ {GΞ,UΞ}.

Proof. Membership carries over – for supported models from (Brewka et al. 2013, Proposition 5), for approximate stable models from Theorem 22. For membership for ultimate stable models, we can use Proposition 28 to adapt the decision procedure of Proposition 21. In any case, hardness carries over from AFs (Dimopoulos and Torres 1996).

For credulous and skeptical reasoning over the two- valued semantics, membership is straightforward and hardness again carries over from argumentation frameworks.

Corollary 32. Let Ξ be a BADF with L=L⁺∪L⁻; consider any operator O ∈ {GΞ,UΞ} and semantics σ∈ {2su,2st}. For s∈S, Cred^O_σ(s) is NP-complete;

Skept^O_σ(s)iscoNP-complete.

Discussion

In this paper we studied the computational complexity of abstract dialectical frameworks using approximation fixpoint theory. We showed numerous novel results for two families of ADF semantics, the approximate and ultimate semantics, which are themselves inspired by argumentation and AFT.

We showed that in most cases the complexity increases by one level of the polynomial hierarchy compared to the corresponding reasoning tasks on AFs. A notable difference between the two families of semantics lies in the stable semantics, where the approximate version is easier than its ultimate counterpart. For the restricted, yet powerful class of bipolar ADFs we proved that for the corresponding reasoning tasks AFs and BADFs have the same complexity, which suggests that many types of relations between arguments can be introduced without increasing the worst-time complexity. On the other hand, our results again emphasize that arbitrary (non-bipolar) ADFs cannot be compiled into equivalent Dung AFs in deterministic polynomial time, unless the polynomial hierarchy collapses to the first level. Under the same assumption, ADFs cannot be implemented directly with methods that are typically applied to AFs, for example answer-set programming (Egly, Gaggl, and Woltran 2010).

Our results lay the foundation for future algorithms and their implementation, for example augmenting the ADF system DIAMOND (Ellmauthaler and Strass 2013) to support also the approximate semantics family, as well as devising efficient methods for the interesting class of BADFs.

For further future work several promising directions are possible. Studying easier fragments of ADFs as well as parameterized complexity analysis can lead to efficient decision procedures, as is witnessed for AFs (Dvoˇr´ak et al.

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2014; Dvoˇr´ak, Ordyniak, and Szeider 2012). We also deem it auspicious to use alternative representations of acceptance conditions, for instance by employing techniques from knowledge compilation (Darwiche and Marquis 2002).

A detailed complexity analysis of other useful AF semantics would also reveal further insights, e.g. semi-stable semantics (Caminada, Carnielli, and Dunne 2012), naive- based semantics, such as cf2 (Baroni, Giacomin, and Guida 2005), or a recently proposed extension-based semantics for ADFs (Polberg, Wallner, and Woltran 2013). For semantical analysis, it would be useful to consider principle-based evaluations for ADFs (Baroni and Giacomin 2007). Furthermore it appears natural to compare (ultimate) ADF semantics and ultimate logic programming semantics (Denecker, Marek, and Truszczy´nski 2004) in approximation fixpoint theory, in particular with respect to computational complexity.

Acknowledgements. Thanks to an anonymous reviewer for pointing out a notational issue in our usage of the complexity classes D^P_i . This research was supported by DFG (project BR 1817/7-1) and FWF (project I1102).

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